论文标题
在非统一的局部驯服奇点上分层的摩尔斯临界点和铜管数
Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularities
论文作者
论文摘要
戈尔斯基（Goresky）和麦克弗森（MacPherson）提出的莫尔斯理论的概括是一个地标，它完全分配了拓扑和地理\ -Me \ -tri \ -tri \ -cal的奇异空间研究。 Let \{$X_t\}_t$ be a suitable family of germs at $0$ of complete intersection varieties in $\mathbb{C}^n$ and $\{f_t\}_t, \{g_t\}_t$ families of non-constant polynomial functions on $X_t$.如果细菌$ x_t $，$ x_t \ cap f_t^{ - 1}（0）$和$ x_t \ cap f_t^{ - 1} { - 1}（0）\ cap g_t^{ - 1}（-1}（0）$是非分类的，local tame tame tame tame，tame tame tame $ t，$ t，$ t，$ t，$ t，$ t，$ t，$ t，$ t，我们的$ b} _ {f_t，x_t}（0）$和$ {\ rm b} _ {f_t，x_t \ cap g_t^{ - 1}（0）}（0）$，与Morse关键点的数量相关$ g_t $具有任意维度的关键基因座。该结果连接拓扑和几何特性，使我们能够确定一些有趣的公式，主要是根据牛顿多面体的组合信息。
The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geo\-me\-tri\-cal study of singular spaces. Let \{$X_t\}_t$ be a suitable family of germs at $0$ of complete intersection varieties in $\mathbb{C}^n$ and $\{f_t\}_t, \{g_t\}_t$ families of non-constant polynomial functions on $X_t$. If the germs $X_t$, $X_t \cap f_t^{-1}(0)$ and $X_t\cap f_t^{-1}(0) \cap g_t^{-1}(0)$ are non-degenerate, locally tame, complete intersection varieties, for each $t,$ we prove that the difference of the Brasselet numbers, ${\rm B}_{f_t,X_t}(0)$ and ${\rm B}_{f_t,X_t\cap g_t^{-1}(0)}(0)$, is related with the number of Morse critical points {on the regular part of the Milnor fiber} of $f_t$ appearing in a morsefication of $g_t$, even in the case where $g_t$ has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.
